Question

Medal + Fan to the person who comes up with the best explanation for d/dx lnx = 1/x!!!!!!!!!!!!

Answer 1

http://www.youtube.com/watch?v=yUpDRpkUhf4

Answer 2

I can't fan KA, though!!!!!!!!!!!!!!!!!!!!!!

Answer 3

The derivative of ln x=1/x, that's the formula, didn't you know that?

Answer 4

Do you know it? Or do you just believe what people tell you?

Answer 5

That's the fact.

Answer 6

Prove it then

Answer 7

Sorry, I don't have time for that and I'm not strong in proofs.

Answer 8

That's like the evolution/creation debate. People believe that we came from a rock 4.6 billion years ago only because people tell them it. Makes no sense!

Answer 9

Who knows what exactly happened 4.6 billion years ago?

Answer 10

So prove that d/dx lnx = 1/x

Answer 11

$$e^{\log x}=x\\\frac{d}{dx}e^{\log x}=\frac{d}{dx}x\\e^{\log x}\frac{d}{dx}\log x=1\\\frac{d}{dx}\log x=\frac1{e^{\log x}}=\frac1x$$

Answer 12

This follows from the proof I gave you for \(\frac{d}{dx}e^x=e^x\) last time :-p

Answer 13

note that to get the third line I applied the chain rule to the left-hand side

Answer 14

\[f(x)=\ln x\\
\begin{align*}\frac{d}{dx}f(x)&=\lim_{h\to0}\frac{\ln(x+h)-\ln x}{h}\\
&=\lim_{h\to0}\frac{\ln\left(\dfrac{x+h}{x}\right)}{h}\\
&=\lim_{h\to0}\frac{\ln\left(1+\dfrac{h}{x}\right)}{h}
\end{align*}\]
Substitute \(t=\frac{h}{x}\) so that \(t\to0\) as \(h\to0\):
\[\begin{align*}\frac{d}{dx}f(x)&=\lim_{t\to0}\frac{\ln\left(1+t\right)}{xt}\\
&=\frac{1}{x}\lim_{t\to0}\frac{\ln\left(1+t\right)}{t}
\end{align*}\]
Next, you can use the fact that \(\ln(1+t)\approx t\) for \(t\) near 0:
\[\begin{align*}\frac{d}{dx}f(x)
&=\frac{1}{x}\lim_{t\to0}\frac{t}{t}\\
&=\frac{1}{x}
\end{align*}\]

Answer 15

@SithsAndGiggles that's a better approach tbh :-p

Answer 16

Gotta use the definition when you can :)